618-322 Topology

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Objectives:

On completion of this subject, students should:

Comprehend:

• the basic concepts and examples of topological spaces;

• the definition of manifolds and the classification of surfaces;

• the idea of homotopy of mappings;

• how to calculate and use the fundamental group;

• the concept of covering spaces and their relationship with fundamental groups;

• the basic ideas of homology theory.

Have developed:

• skills in working with the fundamental group and homology groups;

• the ability to convert problems involving topological spaces and continuous maps into problems in algebra;

• the ability to distinguish between different topological spaces;

• the ability to construct homeomorphisms and homotopy equivalences between spaces.

Appreciate:

• the basic questions in topology;

• the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings;

• how topology can be applied to many areas, including geometry, analysis, group theory and physics.

Content:

Introduction to topology: homotopy and the fundamental group of a space; covering spaces; simplicial homology. Introduction to manifolds: manifolds, tangent vectors, differential forms. Selection of additional topics: connections, Riemannian metrics, curvature, Gauss-Bonnet theorem; integration on manifolds, de Rham's theorem.

Assessment: